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The Thermodynamics of Polymer Mixing and Compression in the Semidilute Regime Nikko Y. Chan and Dave E. Dunstan* Department of Chemical and Biomolecular Engineering, UniVersity of Melbourne, VIC 3010, Australia ReceiVed: February 27, 2010; ReVised Manuscript ReceiVed: June 2, 2010
Polymers present a fascinating range of mechanical properties in the extension, compression, and flow of soft condensed matter. Entropy determines the physics of the elastic material behavior of polymeric systems in deformation. Classical models of polymer dynamics have assumed chain entanglement and resulting reptation in concentrated polymer solutions. Here, we present a thermodynamic treatment of interacting chains in solution with increasing concentration from the dilute to concentrated regimes. As the polymer chain concentration increases above the critical overlap, the chains must either compress or overlap and entangle, resulting in a decrease in chain configurational entropy. The free energy of chain entanglement is shown to be less favored than compression at concentrations above the critical overlap. Elastic forces act on the chains to reduce the dimensions to the ideal random walk size with increasing concentration. At significantly higher concentrations, the free energies reach an asymptote where chain compression and entanglement are simultaneously possible. Entanglement and reptation are shown to be statistically improbable in the semidilute regime, and it is concluded that the compression of polymer chains is favored at semidilute concentrations. Introduction Polymers have been widely studied as a field of condensed matter physics, where the properties of polymers including the structure and fluctuations of molecules, mechanical properties, and reaction kinetics have been studied in detail.1-4 In the 1970s, de Gennes introduced the reptation model to describe the dynamics of polymer chains at high concentrations.5 The chains were assumed to be entangled, and the “ideally mixed” solutions were restricted to “worm-like” reptative motion of the chains. The assumption that the chains are restricted to a tube by the presence of the other chains led to the D ∼ M-2 result for the predicted diffusion of the chains with molecular weight. Considerable effort has been undertaken to measure the molecular weight dependence of the self-diffusion coefficient with varying degrees of success, as reviewed by Lodge.6 Entanglement of polymer molecules in solution is generally assumed in the field, leading to evolved theories of reptation for polymers7 and more recent reptation models in order to model melt rheology.8,9 However, to date, no direct observation of entanglements has been made. Numerous studies of polymer physics and dynamics have shown contrasting results, including studies of DNA molecules in shear flow10-13 and studies of synthetic polymer systems.14-16 Both sets of experimental studies show that the polymer chains tumble in flow at concentrations above the critical overlap. It is difficult to reconcile chain tumbling and compression with ideally mixed and entangled systems. However, the time dependent relaxation of the moduli measured for polymer melts is strong evidence for entanglement.17 In order to further understand these observations, the thermodynamics of polymer molecules in solutions of good solvent has been investigated. Theory
cal mechanics.1,4 The polymer chain is treated as a random walk on a periodic lattice where the essential physics of the chain is determined by the configurational entropy. The introduction of scaling arguments established universal applicability of the theory whereby the chain backbone chemistry may be neglected.1,4 Creative use of scaling arguments and statistical mechanics has enabled definition of chain dimensions in the quiescent state which has been experimentally verified using light scattering.1 Self-avoiding “real” random walk models yield an elegantly simple model of these complex systems where the chain size and molecular weight relationship are described as proposed by Flory4
RF ∝ N3/5
(1)
where RF is the end-to-end distance of the self-avoiding random walk or the Flory radius and N is the number of segments which are randomly oriented in space. The N segments are independent of bond angle chemistry and are therefore random walks providing N is large.1 Equation 1 is derived from an energy minimization argument first proposed by Flory.3 Figure 1 shows the minimization between the elastic entropy due to chain freedom and interactions due to excluded volume, as given by eqs 2 and 3.2,4 Excluded volume acts to expand the chain, while the elastic contribution resists the expansion. The minimum in the free energy occurs at an expanded polymer size compared to the random walk as a result of these opposing contributions.2,17 Equation 1 has been shown by a number of methods to be a realistic description of the measured chain sizes in good solutions.1 Method
A significant development in theoretical physics has been to describe the solution behavior of macromolecules using statisti* To whom correspondence should be addressed. E-mail: davided@ unimelb.edu.au.
Herein, the equations describing the elastic and mixing contributions to the free energy are used to interpret the interaction energy between chains with increasing polymer concentration.
10.1021/jp101793c 2010 American Chemical Society Published on Web 07/28/2010
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J. Phys. Chem. B, Vol. 114, No. 32, 2010 10521
r0 ) N1/2a
(5)
Equation 4 is valid for both chain stretching as has been assumed in shear flow and compression as for rubber elasticity as a result of doing work on the chain via an imposed stress.18 There are two scenarios to describe the behavior of polymer chains in solution as the concentration increases above critical overlap, where either the chains must compress and reduce the occupied volume per chain or the chains must mix and entangle. For a purely entanglement case with no change in the polymer end-to-end distance, the entropy decreases as the chains begin to interpenetrate, causing an increase in the interaction energy. The dependence of the free energy with increasing concentration is obtained by the increase in the effective N number of segments in the given volume. The increase in the free energy per chain for chain mixing is then proportional to the increase in concentration of polymer molecules in solution as given by Figure 1. Plot of free energy minimum between the combined effect of the entropy of elasticity and the excluded volume interaction to derive the Flory result.
The free energy associated with the entropy change due to elastic deformation for the ideal freely jointed chain or so-called phantom chain has previously been derived by Flory as2,3,17
Fel )
3kBTr2 2Na2
(2)
where kB is Boltzmann’s constant, T is the temperature, r is the root-mean-square end-to-end distance of the polymer molecule, and a is the length of the freely jointed polymer segment. The repulsive free energy due to the change in entropy of two chains mixing, using excluded volume arguments, is given by1,2,4,17
Frep )
4kBTυN2 r3
Fmix )
4kBTυN2(c/c*) RF3
(6)
where c is the concentration of polymer and c* the critical overlap concentration. As the concentration of polymer molecules is increased beyond c*, the number of polymer segments in the same volume increases proportionally. The use of this equation yields an expression for the repulsive free energy experienced by polymer chains mixing and subsequent entanglement. The kinetics of chain interpenetration and attainment of equilibrium mixing are assumed instantaneous and are not considered further here. Likewise, for a case where the polymer molecules deform and do not overlap, the end-to-end distance r for each molecule will be reduced while the number of segments N remains constant in a reduced volume. The resulting change in repulsive free energy associated with deformation of individual molecules is as given in eq 3. The reduction in end-to-end distance will
(3)
where υ is the excluded volume of the polymer chain segments. These equations are extended to the case of many chains in a finite volume at thermal equilibrium with complete mixing and therefore isotropic. The initial end-to-end distance of the polymer, when the concentration is at or below c*, is assumed to be the Flory radius RF. Elastic forces act on the polymer chain to resist deviation from the ideal random walk dimensions through compression or extension. The chain entropy is at a maximum when the polymer adopts the random walk conformation. The elastic free energy of an ideal polymer chain with changing end-to-end distance is described by eq 4 as a deviation from the ideal chain size, r0, as derived using random walk statistics. Figure 2 shows the difference in equilibrium chain size comparing ideal chains which only have the elastic contribution and real chains which have both elastic and excluded volume effects.
Fel )
3kBT(r - r0)2 2Na2
(4)
Figure 2. Plot of normalized free energy for changes in entropy due to elastic deformation. The maximum entropy state for an ideal chain occurs at r0 corresponding to the ideal random walk, while real chains settle at RF due to excluded volume effects.
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Figure 3. Plot of normalized free energy change for deviation from RF for polymer mixing compared to elastic deformation as a function of polymer size.
be geometrically related to the increase in polymer concentration, assuming hard spheres, as
( )
c r ) c* RF
-3
(7)
energy change for deformation is large, indicating that it would be unfavorable for the molecule to extend beyond RF. Discussion
Substitution of eq 7 into eq 3 will give the same functional form as eq 4, which shows that the change in repulsive free energy due to an increasing concentration of overlapping chains is identical to an equivalent decrease in polymer size. The free energy for the elastic deformation of molecules will be defined as the sum of elastic and repulsive excluded volume interactions, given as Fdef in eq 8. The calculation of Fel is based on the ideal chain; therefore, the deviation from ideality due to excluded volume is accounted for by the addition of the repulsive free energy term Frep.
Fdef ) Fel + Frep
Figure 4. Plot of normalized free energy change for deviation from RF for polymer mixing compared to elastic deformation as a function of concentration on the logarithmic scale. The lower free energy for deformation above c* shows that this is the favored state.
(8)
Results The free energy changes for polymer mixing and for polymer compression are plotted in Figures 3 and 4 for a typical polymer system with 2000 segments. The point at which c/c* ) 1 is taken to be r/RF ) 1, and the free energy changes are calculated for a deviation from RF. For the typical polymer system at concentrations above c*, the free energy change required for molecular deformation is less than mixing; therefore, entanglements are improbable. The reason that the free energy for deformation is less than mixing is that the elastic force acts as a restorative force to compress the chains from RF to r0, with the driving force being the deviation from the ideal chain size r0. As such, Fel is a negative contribution to Fdef with a minimum at r ) r0 where the ideal chain is realized. At concentrations well above c*, the free energy changes for deformation and mixing approach the same value, indicating that chain mixing and entanglement become thermodynamically possible and will occur simultaneously with chain compression. When this situation occurs, the size of the polymer chains approaches the ideal random walk dimensions as proposed by Flory. For concentrations below c*, the free
The concept of entanglement in polymer physics for concentrated solutions and melts is generally accepted in the field.18-21 To date, however, no direct evidence exists to show that entanglements exist in semidilute solutions. A body of literature pertaining to the rheologically measured plateau modulus and interpretation of the relaxation time exists,19,20 and has been interpreted assuming the concept of entanglements, in order to predict the observed frequency dependence of the storage modulus.19,20 In these models, the relaxation times of the moduli are interpreted using an effective tube relaxation time.17 Modifications to the standard models using constraint release have been proposed to fit the rheological data for a range of molecular weights and polymers in flow.22 A significant body of literature assumes entanglements exist in order to model the observed rheological behavior.21 The theory of rubber elasticity assumes that both entanglements and cross-links exist in order to predict the observed behavior.18 Recent experimental evidence from rheofluorescence measurements,14-16,23 fluorescence anisotropy,24 and confocal measurements on DNA in flow10 have suggested that polymer chain compression or tumbling are occurring in simple flow. It would be highly improbable for fully entangled chains to tumble in simple shear flow, and even less likely for compression to occur. This shows that the concept of entanglement is not universal for all polymer systems at concentrations above c*. Polymer chains would be expected to be entangled in polymer melts, but as the chains are exposed to solvent, the predicted behavior would be disentanglement given sufficient dilution and time to reach equilibrium. Further dilution would result in swelling of the chain until RF is achieved at a concentration equal to or below c*. These results are in accord with the thermodynamic findings of the present work. Indeed, the thermodynamic treatment presented here indicates that modeling polymer flow behavior in terms of “compressed blobs” rather that entangled chains is required.
Thermodynamics of Polymer Mixing and Compression There have been several studies employing techniques such as neutron and X-ray scattering to probe the polymer chain sizes from dilute to concentrated regimes.25-30 These have been clearly summarized by Graessley.21 These studies have shown a range of inconsistent results regarding chain size with concentration in the semidilute regime. These attempts to verify the theories are to date inconclusive. In work by Richter and co-workers, the reptation model was shown to be incorrect in the fitting of neutron scattering data of PDMS samples ranging in concentration from the dilute to concentrated.26 More recent work by Richter et al. appears to be consistent with the predictions of scaling theory.27 Conclusions We conclude that, for real chains, the free energy of compression becomes less than the free energy of mixing when the polymer concentration is increased beyond critical overlap. The behavior predicted by this model shows that elastic compression of the chains is favored. At much higher concentrations and in polymer melts, the free energies reach an asymptote and the Flory result becomes valid; however, the excluded volume is still finite under these conditions. Traditional models for the behavior of polymer molecules in solution have generally assumed that molecules favor entanglement and reptation. The thermodynamic treatment presented here shows that entanglement is thermodynamically unfavorable, resulting in the improbability of reptation occurring in these regimes. Models in the foundation of polymer physics and study of soft condensed matter must be addressed. References and Notes (1) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (2) Teraoka, I. Polymer Solutions. An Introduction to Physical Properties; Wiley-Interscience: New York, 2002. (3) Flory, P. J. Statistical Mechanics of Chain Molecules; Hanser Publications: New York, 1988.
J. Phys. Chem. B, Vol. 114, No. 32, 2010 10523 (4) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (5) de Gennes, P. G. J. Chem. Phys. 1971, 55, 572. (6) Lodge, T. P. Phys. ReV. Lett. 1999, 83, 3218. (7) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon Press: Oxford, U.K., 1986. (8) Marrucci, G.; Hermans, J. J. Macromolecules 1980, 13, 380. (9) Marrucci, G.; Grizzuti, N. J. Non-Newtonian Fluid Mech. 1986, 21, 319. (10) LeDuc, P.; Haber, C.; Bao, G.; Wirtz, D. Nature 1999, 399, 564. (11) Doyle, P. S.; Ladoux, B.; Viovy, J. L. Phys. ReV. Lett. 2000, 84, 4769. (12) Babcock, H. P.; Smith, D. E.; Hur, J. S.; Shaqfeh, E. S. G.; Chu, S. Phys. ReV. Lett. 2000, 85, 2018. (13) Smith, D. E.; Babcock, H. P.; Chu, S. Science 1999, 283, 1724. (14) Dunstan, D. E.; Hill, E. K.; Wei, Y. Macromolecules 2004, 37, 1663. (15) Dunstan, D. E. Eur. J. Phys. 2008, 29, 977. (16) Chan, N. Y.; Chen, M.; Dunstan, D. E. Eur. Phys. J. E 2009, 30, 37. (17) Jones, R. A. L. Soft Condensed Matter; Oxford University Press: New York, 2002. (18) Treloar, L. R. G. The Physics of Rubber Elasticity, 3rd ed.; Oxford University Press: Oxford, U.K., 2005. (19) Pattamaprom, C.; Larson, R. G. Macromolecules 2001, 34, 5229. (20) Park, S. J.; Larson, R. G. J. Rheol. 2003, 47, 199. (21) Graessley, W. W. Polymeric Liquids & Networks: Structure and Properties; Garland Science: New York, 2004. (22) Graessley, W. W. AdV. Polym. Sci. 1982, 47, 68. (23) Dunstan, D. E.; Wei, Y. Eur. Phys. J.: Appl. Phys. 2007, 38, 93. (24) Bur, A. J.; Lowry, R. E.; Roth, S. C.; Thomas, C. L.; Wang, F. W. Macromolecules 1991, 24, 3715. (25) Westermann, S.; Willner, L.; Richter, D.; Fetters, L. J. Macromol. Chem. Phys. 2001, 201, 500. (26) Richter, D.; Binder, K.; Ewen, B.; Stu¨hn, B. J. Phys. Chem. 1984, 88, 6618. (27) Zamponi, M.; Wischnewski, A.; Monkenbusch, M.; Willner, L.; Richter, D.; Falus, P.; Farago, B.; Guenza, M. G. J. Phys. Chem. B 2008, 112, 16220. (28) Arbe, A.; Monkenbusch, M.; Stellbrink, J.; Richter, D.; Farago, B.; Almdal, K.; Faust, R. Macromolecules 2001, 34, 1281. (29) Daoud, M.; Cotton, J. P.; Farnoux, B.; Jannink, G.; Sarma, G.; Benoit, H.; Duplessix, R.; Picot, C.; de Gennes, P. G. Macromolecules 1975, 8, 804. (30) Kent, M. S.; Tirrell, M.; Lodge, T. P. Polymer 1991, 32, 314.
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